In a sense, you are right: if we generate forecasts for the 0.001, 0.002, ..., 0.998 and 0.999 quantile, then we pretty much already have a full probabilistic forecast. Essentially, the predicted density would be a histogram with 998 bins.
However, I have rarely seen this. (One of the rare examples is the GEFCom2014 competition; Hong et al., 2016, IJF, which required submitting 99 quantile forecasts.) More frequently, one sees people doing it the other way around: predicting a density and deriving quantile forecasts from that.
One potential problem is that quantile forecasts for very close quantiles may be inconsistent: the 0.998 quantile forecast should always be lower than the 0.999 quantile forecast, but if you don't take particular care, it may be the other way around for some time points in the future. This problem also afflicts quantile regression and prediction. Of course, this problem will be more prevalent if your quantiles are close together.
Incidentally, and just to help search engines, related terms are density forecasting and predictive densities or predictive distributions (the latter being the output from the former).